Problem: Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{t^2 - 2t - 24}{t^2 + 3t - 54}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 - 2t - 24}{t^2 + 3t - 54} = \dfrac{(t + 4)(t - 6)}{(t + 9)(t - 6)} $ Notice that the term $(t - 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t - 6)$ gives: $z = \dfrac{t + 4}{t + 9}$ Since we divided by $(t - 6)$, $t \neq 6$. $z = \dfrac{t + 4}{t + 9}; \space t \neq 6$